Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

Options

A. f(x)
B. f'(x)
C. f''(x)
D. none of these

Answer

To solve the problem, we need to show that if \( a \) and \( b \) are two distinct roots of a polynomial equation \( f(x) = 0 \), then there exists at least one root lying between \( a \) and \( b \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a polynomial \( f(x) \) such that \( f(a) = 0 \) and \( f(b) = 0 \), where \( a \) and \( b \) are distinct roots. We need to prove that there exists at least one root \( c \) in the interval \( (a, b) \).

**Hint**: Identify the properties of polynomial functions, particularly their continuity and differentiability.

2. **Applying the Intermediate Value Theorem**:
Since \( f(x) \) is a polynomial, it is continuous everywhere on \( \mathbb{R} \). Because \( f(a) = 0 \) and \( f(b) = 0 \), we can analyze the values of \( f(x) \) at points in the interval \( (a, b) \).

**Hint**: Recall that the Intermediate Value Theorem states that if a function is continuous on an interval and takes on two different values at the endpoints, it must take on every value in between.

3. **Analyzing the Sign of \( f(x) \)**:
Since \( f(a) = 0 \) and \( f(b) = 0 \), we can consider the behavior of \( f(x) \) in the interval \( (a, b) \). If \( f(x) \) changes sign between \( a \) and \( b \), then by the Intermediate Value Theorem, there must be some \( c \) in \( (a, b) \) such that \( f(c) = 0 \).

**Hint**: Think about what happens if \( f(x) \) is positive at one endpoint and negative at the other.

4. **Using Rolle's Theorem**:
Since \( f(x) \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \), we can apply Rolle's Theorem. This theorem states that there exists at least one point \( c \) in \( (a, b) \) such that \( f'(c) = 0 \).

**Hint**: Remember that if a function has equal values at two points, its derivative must be zero at some point in between.

5. **Conclusion**:
The existence of a point \( c \) such that \( f'(c) = 0 \) indicates that there is at least one root of the derivative \( f'(x) \) in the interval \( (a, b) \). Since \( f'(x) \) is also a polynomial, it will have roots, and thus there is at least one root of \( f(x) \) in the interval \( (a, b) \).

**Hint**: Conclude by summarizing how the continuity and differentiability of polynomials lead to the existence of roots.

### Final Statement:
Thus, we have shown that if \( a \) and \( b \) are distinct roots of the polynomial equation \( f(x) = 0 \), then there exists at least one root lying between \( a \) and \( b \).